Hypergeometric3F2Regularized

The regularized generalized hypergeometric function ${}_3F_2(a_1,a_2,a_3;b_1,b_2;z) / (\Gamma(b_1)\Gamma(b_2))$ — entire in $b_1, b_2$; only converges for $|z| < 1$. Provided by @enumeratio/analytic.

FungrimHypergeometric3F2Regularized↗Fungrim entry6582c4↗e1797b↗

Hypergeometric3F2Regularized(a1, a2, a3, b1, b2, z)${}_3F_2(a_1,a_2,a_3;b_1,b_2;z) / (\Gamma(b_1)\Gamma(b_2))$.

SymPyhyper([a1, a2, a3], [b1, b2], z) / (gamma(b1) * gamma(b2))mpmathenumeratio_hyperreg([a1, a2, a3], [b1, b2], z)

Typed signatures

Type	Package	Applies
(number, number, number, number, number, number) -> number	analytic	always

Domain: Special functions

Details

• compute-engine has no Hypergeometric3F2 at all, regularized or otherwise. Computed directly by the $1/\Gamma$-per-term series ($p = q + 1$ here), finite at either $b_1$ or $b_2$ a nonpositive integer.
• The series only converges for $|z| < 1$; outside the unit disc this stays symbolic. N(…, d) past a double's digits runs a bignum series for real arguments with $|z| < 1$ (the cancellation at negative $z$ is paid for in extra working digits), and stays symbolic rather than padding a double where it can't settle: a complex $z$, $|z| \ge 1$, or $z$ so near the rim that the series outruns its term budget. Fungrim's own identities for this head (e.g. the Chebyshev derivative formulas, fungrim:6582c4 / fungrim:e1797b) are unconstrained in their own argument, so not every instance evaluates.
• Wolfram has no dedicated 3,2 head; it maps to the generic HypergeometricPFQRegularized[{a1,a2,a3},{b1,b2},z].

Plot

Examples

9 more kept as data, checked against the oracles

Basic #

a generic point inside the unit disc

live ↗

In$\mathrm{Hypergeometric3F2Regularized}(1, 1, 1, 2, 3, 0.5)$Out$0.550\,775\,414\,049\,914\,4$

Scope #

live ↗

In$N(\mathrm{Hypergeometric3F2Regularized}(1, 1, 1, 2, 3, \frac{1}{2}), 50)$Out$0.550\,775\,414\,049\,915\,630\,639\,776\,883\,235\,713\,353\,639\,397\,218\,332\,76$

Possible issues #

$|z| \ge 1$ stays symbolic — no continuation past the unit disc

live ↗

In$\mathrm{Hypergeometric3F2Regularized}(1, 1, 1, 2, 3, 1.2)$Out$\mathrm{Hypergeometric3F2Regularized}(1, 1, 1, 2, 3, 1.2)$

live ↗

In$N(\mathrm{Hypergeometric3F2Regularized}(1, 1, 1, 2, 3, \frac{6}{5}), 30)$Out$\mathrm{Hypergeometric3F2Regularized}(1, 1, 1, 2, 3, 1.2)$

live ↗

In$N(\mathrm{Hypergeometric3F2Regularized}(1, 2, 3, 4, 5, \frac{999}{1\,000}), 50)$Out$\mathrm{Hypergeometric3F2Regularized}(1, 2, 3, 4, 5, 0.999)$

About

Examples	14
Checked against	Wolfram Language 11 agree3 differSymPy 5 agree6 differ3 unevaluatedmpmath 11 agree2 differ1 error
Crosswalk	FungrimFungrim entry ×2SymPympmath

Implementation

primitive · numericirreducibly numeric — it evaluates rather than rewrites

nativeenginetypescriptpackages/symbols/analysis/analytic/src/hypergeometric.ts

mappedexternalwolfram / mpmathpackages/oracle/src/mappings.ts

HypergeometricPFQRegularized[{a1,a2,a3},{b1,b2},z]; mpmath's own regularized series (rgamma per term).

mappedmpmathpackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

SymPy has no regularized hypergeometric, so this divides by Γ(b1)Γ(b2); at a pole of either (a non-positive integer) the quotient is not the limit that ours is.

See also: Hypergeometric2F1Regularized